# proof of ratio test

Assume $$. By definition $\exists N$ such that

$$

i.e. eventually the series $|{a}_{n}|$ becomes less than a convergent^{} geometric series^{}, therefore a shifted subsequence^{} of $|{a}_{n}|$ converges by the comparison test^{}. Note that a general sequence^{} ${b}_{n}$ converges iff a shifted subsequence of ${b}_{n}$ converges. Therefore, by the absolute convergence theorem, the series ${a}_{n}$ converges.

Similarly for $k>1$ a shifted subsequence of $|{a}_{n}|$ becomes greater than a geometric series tending to $\mathrm{\infty}$, and so also tends to $\mathrm{\infty}$. Therefore ${a}_{n}$ diverges.

Title | proof of ratio test |
---|---|

Canonical name | ProofOfRatioTest |

Date of creation | 2013-03-22 12:24:46 |

Last modified on | 2013-03-22 12:24:46 |

Owner | vitriol (148) |

Last modified by | vitriol (148) |

Numerical id | 6 |

Author | vitriol (148) |

Entry type | Proof |

Classification | msc 40A05 |

Classification | msc 26A06 |